The generalized feedback shift register (GFSR) algorithm suggested by Lewis and Payne is a widely used pseudorandom number generator, but has the following serious drawbacks: (1) an initialization scheme to assure higher order equidistribution is involved and is time consuming(2) each bit of the generated words constitutes an m-sequence based on a primitive trinomial, which shows poor randomness with respect to weight distribution;(3) a large working area is necessary (4) the period of sequence is far shorter than the theoretical upper bound. This paper presents the twisted GIFSR (TGFSR) algorithm, a slightly but essentially modified version of the GFSR, which solves all the above problems without loss of merit. Some practical TGFSR generators were implemented and passed strict empirical tests. These new generators are most suitable for simulation of a large distributive system, which requires a number of mutually independent pseudorandom number generators with compact size.

The twisted GFSR generators proposed in a previous article have a defect in k -distribution for k larger than the order of recurrence. In this follow up article, we introduce and analyze a new TGFSR variant having better k -distribution property. We provide an efficient algorithm to obtain the order of equidistribution, together with a tight upper bound on the order. We discuss a method to search for generators attaining this bound, and we list some of these such generators. The upper bound turns out to be (sometimes far) less than the maximum order of equidistribution for a generator of that period length, but far more than that for a GFSR with a working are of the same size.

The fixed vector of any m-sequence based on a trinomial is explicitly obtained. Local nonrandomness around the fixed vector is analyzed through model-construction and experiments. We conclude that the initial vector near the fixed vector should be avoided.

All primitive trinomials over GF (2) with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are X 859433 + X 288477 + 1 and its reciprocal. Also two examples of primitive pentanomials over GF (2) with degree 86243 (which is the 28th Mersenne exponent) are presented. The sieve used is briefly described.

Abstract. All of the primitive trinomials over GF(2) with degree p given by one of the Mersenne exponents 19937, 21701, 23209, and 44497 are presented. Also, one example of a primitive pentanomial over GF(2) is presented for each degree up to 44497 that is a Mersenne exponent. The sieve used is briefly described. A problem is posed which conjectures the number of primitive pentanomials of degree p .